Studying Micro to Millisecond Protein Dynamics Using Simple Amide ¹⁵N CEST Experiments Supplemented with Major-State R₂ and Visible Peak-position Constraints

Abstract

Over the last decade amide ¹⁵N CEST experiments have emerged as a popular tool to study protein dynamics that involves exchange between a ‘visible’ major state and sparsely populated ‘invisible’ minor states. Although initially introduced to study exchange between states that are in slow exchange with each other (typical exchange rates of, ~10 to ~400 s^-1), they are now used to study interconversion between states on the intermediate to fast exchange timescale while still using low to moderate (~5 to ~300 Hz) ‘saturating’ B₁ fields. The ¹⁵N CEST experiment is very sensitive to exchange as the exchange delay T_EX can be quite long (~0.5 s) allowing for a large number of exchange events to occur making it a very powerful tool to detect minor sates populated (p_minor) to as low as 1%. When systems are in fast exchange and the ¹⁵N CEST experiments readily detect the minor states, the exchange parameters are often still poorly defined because the ${\chi }_{red}^2$ versus p_minor and ${\chi }_{red}^2$ versus exchange rate (k_ex) plots can be quite flat with shallow or no minima and the analysis of such ¹⁵N CEST data can lead to wrong estimates of the exchange parameters due to the presence of ‘spurious’ minima. Here we show that the inclusion of experimentally derived constraints on the intrinsic transverse relaxation rates and the inclusion of visible state peak-positions during the analysis of amide ¹⁵N CEST data acquired with moderate B₁ values (~50 to ~300 Hz) results in a convincing minimum in the ${\chi }_{red}^2$ versus p_minor and the ${\chi }_{red}^2$ versus k_ex plots even when exchange occurs on the ~100 μs timescale. The utility of this strategy is demonstrated on the fast-folding Bacillus stearothermophilus peripheral subunit binding domain that folds with a rate constant ~10,000 s^-1. Here the analysis of ¹⁵N CEST data alone results in ${\chi }_{red}^2$ versus p_minor and ${\chi }_{red}^2$ versus k_ex plots that contain shallow minima, but the inclusion of visible-state peak positions and restraints on the intrinsic transverse relaxation rates of both states during the analysis of the ¹⁵N CEST data results in pronounced minima in the ${\chi }_{red}^2$ versus p_minor and ${\chi }_{red}^2$ versus k_ex plots and precise exchange parameters even in the fast exchange regime k_ex/|Δω| ~5). Using this strategy we find that the folding rate constant of PSBD is invariant (~10,500 s^-1) from 33.2 to 42.9 °C while the unfolding rate (~70 to ~500 s^-1) and unfolded state population (~0.7 to ~4.3%) increase with temperature. The results presented here show that protein dynamics occurring on the ~10 to ~10,000 s^-1 timescale can be studied using amide ¹⁵N CEST experiments.

Introduction

Protein molecules interconvert among a wide array of conformational states over a broad range of timescales varying from picoseconds to seconds (Bahar, Jernigan, and Dill, 2017; Karplus, 2000). Conformational dynamics occurring over the μs to second time-scale often involves a dominant major conformational state that exchanges with various sparsely populated states. Owing to their low populations and short lifetimes these sparsely populated minor states are not visible in standard NMR spectra that contain signals only from the dominant major state (Cavanagh et al., 2006). Hence the sparsely populated states are referred to as ‘invisible’ states while the major state is termed the ‘visible’ state. As these sparsely populated invisible states play crucial roles in protein function, folding, misfolding and aggregation (Bahar, Jernigan, and Dill, 2017; Milojevic et al., 2007; Sekhar and Kay, 2019), various NMR experiments that manipulate the visible state magnetisation to detect these ‘invisible’ states have been developed over the past three decades (Anthis and Clore, 2015; Palmer and Koss, 2019; Rangadurai et al., 2019; Sekhar and Kay, 2019; Torchia, 2011; Tugarinov and Clore, 2019; Vallurupalli et al., 2017; Zhuravleva and Korzhnev, 2017). These include the R_1ρ (Palmer and Massi, 2006; Rangadurai et al., 2019), CPMG (Carr-Purcell-Meiboom-Gill) (Hansen, Vallurupalli, and Kay, 2008; Loria, Rance, and Palmer, 1999a; Palmer, Kroenke, and Loria, 2001), DEST (Dark-state Exchange Saturation Transfer) (Fawzi et al., 2011; Tugarinov and Clore, 2019) and CEST (Chemical Exchange Saturation Transfer) (Forsen and Hoffman, 1963; Vallurupalli, Bouvignies, and Kay, 2012; Ward, Aletras, and Balaban, 2000) classes of experiments. The R_1ρ, CPMG and CEST class experiments detect ‘invisible’ states based on differences in the chemical shifts (Allerhand and Thiele, 1966) between the major and minor states, while the DEST methodology exploits differences in the transverse relaxation of rates (Allerhand and Thiele, 1966) using the visible major state peak to detect ‘dark’ states that have very large transverse relaxation rates (Fawzi et al., 2011; Tugarinov and Clore, 2019).

The CEST class of experiments that were first described by Forsen and Hoffman sixty years ago to study chemical exchange between visible states (Forsen and Hoffman, 1963) have subsequently been used in imaging (van Zijl and Yadav, 2011; Ward, Aletras, and Balaban, 2000) and to study exchange between a visible major state and sparsely populated minor state(s) (Vallurupalli, Bouvignies, and Kay, 2012; Vallurupalli et al., 2017; Zhao, Baisden, and Zhang, 2020). In a typical CEST experiment used to study exchange between a visible major state (A) and an invisible minor state (B) that are in slow exchange $(A\underset{k_{AB}}{\stackrel{k_{BA}}{\leftrightharpoons }}B)$ with each other, longitudinal magnetisation is subjected to a weak B₁ field (~5 to ~50 Hz) for a time exchange time T_EX (~300 to 600 ms) and the intensity (I) of the visible peak is quantified as a function of the offset at which the B₁ field is applied (Palmer and Koss, 2019; Sekhar and Kay, 2019; Vallurupalli et al., 2017). A plot of the normalised intensity (I/I₀) of the visible state versus the offset (ϖ_RF) at which the B₁ field is applied is called the CEST intensity profile and will contain dips at the chemical shifts of both the major state A (ϖ_A) and the minor state (ϖ_B). Here I₀ is the intensity of the visible state in the absence of the T_EX delay. The exchange rate (k_ex,AB = k_AB + k_BA), the fractional population of the minor state (p_B = k_AB/k_ex,AB), the chemical shift (ϖ_B) and in favourable cases the transverse relaxation rate (R_2,B) of the minor state can all be obtained by analysing the CEST intensity profiles recorded at two B₁ fields leading to a complete description of the exchange processes and allowing one to reconstruct the spectrum of the ‘invisible’ state. ¹⁵N and ¹³C CEST experiments were initially used to detect minor states (Bouvignies and Kay, 2012; Bouvignies, Vallurupalli, and Kay, 2014; Vallurupalli, Bouvignies, and Kay, 2012) that are in slow exchange (k_ex,AB/|Δω_AB| < ~0.5) with the major state, so that separate dips corresponding to major and minor states could be observed in the intensity profiles of at least a few residues. Here, ω_A and ω_B are the resonance frequencies (rad/s) of the spin of interest in states A and B respectively and Δω_AB = ω_B – ω_A. Similarly (in ppm) we have Δϖ_AB = ϖ_B – ϖ_A. Due to the limited range of |Δω_AB| values in protein samples, this requirement for separate major and minor state dips in the CEST intensity profiles limited the applicability of the CEST experiments to protein exchange processes with exchange rates less than ~400 s^-1, while CPMG and R_1ρ experiments were used to study exchange occurring on the micro to millisecond timescale (Massi et al., 2004; Palmer, Kroenke, and Loria, 2001; Sekhar and Kay, 2013; Zhuravleva and Korzhnev, 2017). When exchange between states A and B lies in the intermediate to fast exchange limit (~0.7 < k_ex,AB/|Δω_AB| < ~5) the CEST intensity profile will not contain two distinct dips arising from states A and B, but will contain a single asymmetric dip (Rangadurai, Shi, and Al-Hashimi, 2020). Just as off-resonance R_1ρ data (Palmer and Massi, 2006; Rangadurai et al., 2019) has been analysed to study exchange occurring on the intermediate to fast exchange timescale, these asymmetric CEST intensity profiles obtained with larger B₁ values (~100 to ~300 Hz) that lack distinct dips can also be analysed to obtain the exchange parameters (k_ex,AB and p_B), minor state chemical shifts etc (Avram et al., 2017; Ramanujam, Charlier, and Bax, 2019; Rangadurai, Shi, and Al-Hashimi, 2020; Tiwari et al., 2021). A particularly pleasing aspect of these CEST experiments is the use of very modest B₁ fields (< ~300 Hz) that are not taxing on the probe to characterise exchange processes occurring at rates on the order of 10,000 s^-1. In contrast to R_1ρ experiments (Korzhnev, Orekhov, and Kay, 2005; Massi et al., 2004), in the case of CEST experiments there is no need to use intricate schemes to align the magnetisation along the applied B₁ field allowing one to easily use small to moderate B₁ fields at any desired offset and easily collect intensity profiles over a wide chemical shift range. As the T_EX delay in a CEST experiment is usually quite long (~0.5 s), several exchange events occur during T_EX making them very sensitive to exchange and because the CEST experiments often do not need specially labelled samples, they have been used to study several processes including protein folding, misfolding & aggregation (Goerke et al., 2017; Lim et al., 2014; Sekhar et al., 2015; Tiwari et al., 2021), protein/nucleic acid conformational exchange (Deshmukh et al., 2016; Gladkova et al., 2017; Rangadurai, Shi, and Al-Hashimi, 2020; Zhao et al., 2017), reaction mechanisms (Ramanujam, Charlier, and Bax, 2019; Sekhar et al., 2018) etc to name a few. Consequently new CEST experiments and strategies are continuously being developed, including ones that extend the applicability of CEST experiments to study exchange between multiple states (Tiwari et al., 2021; Vallurupalli, Tiwari, and Ghosh, 2019), to probe new sites in molecules (Karunanithy, Reinstein, and Hansen, 2020; Pritchard and Hansen, 2019; Tiwari and Vallurupalli, 2020; Yuwen and Kay, 2018; Yuwen, Sekhar, and Kay, 2017), to expedite data analysis (Chao, Zhang, and Byrd, 2021; Karunanithy et al., 2022), to expedite acquisition (Bolik-Coulon, Hansen, and Kay, 2022; Jameson et al., 2019; Leninger et al., 2018; Toyama and Shimada, 2019; Yuwen, Bouvignies, and Kay, 2018; Yuwen, Kay, and Bouvignies, 2018), to deal with artefacts (Tiwari, Pandit, and Vallurupalli, 2019; Xia et al., 2021) etc.

Here we have used ¹⁵N CEST experiments with B₁ values ranging from ~50 to ~300 Hz to study the folding of the ~4.7 kDa peripheral subunit binding domain (PSBD) from the pyruvate dehydrogenase multienzyme complex of Bacillus stearothermophilus that folds on the ~100 μs timescale (Vugmeyster et al., 2000). Under conditions used here, at 42.9 °C the unfolded (U) state has a population (p_U) of ~4.3% and the exchange rate (k_ex,FU) between the folded (F) and unfolded (U) state is ~11,739 s^-1. The ¹⁵N CEST intensity profiles were asymmetric and incompatible with a one-state system but are compatible with a two-state (F⇌U) exchange model in which the protein exchanges between the native folded state (F) and the unfolded state (U). Although some ¹⁵N |Δϖ_FU| values were as large as ~8 ppm (~3566 rad/s at 700 MHz), the two-state ${\chi }_{red}^2$ versus p_U and ${\chi }_{red}^2$ versus k_ex,FU plots are quite flat and when ¹⁵N CEST data was analysed only from a restricted set of residues for which |Δϖ_FU| < 5ppm, the ${\chi }_{red}^2$ versus p_U plot does not contain a convincing minimum. Here we show that the analysis of the ¹⁵N CEST data along with experimentally derived restraints on the folded state intrinsic (exchange-free) transverse relaxation rate (R_2,F) and visible state peak-positions (ϖ_Vis) resulted in sharper and more pronounced minima in the ${\chi }_{red}^2$ versus p_U and ${\chi }_{red}^2$ versus k_ex,FU plots even when only residues with |Δϖ_FU| < 5ppm (|Δω_FU | < ~2230 rad/s at 700 MHz) were analysed showing that ¹⁵N CEST data supplemented with experimentally derived restraints can be used to study relatively fast processes with k_ex/|Δω| ~5. The folding kinetics and thermodynamics of PSBD were studied as a function of temperature (33.2 to 42.9 °C). As with other fast folding proteins the folding rate of PSBD is essentially invariant with temperature while the unfolding rate increases with temperature. An Arrhenius analysis of the temperature dependent rates suggests that PSBD folds over a small barrier involving a transition state that is more ordered and contains more energetically favourable interactions than the U state.

Materials and methods

Sample details

U-[¹⁵N] PSBD was expressed in BL21(DE3) E. coli cells and purified as described previously (Gopalan and Vallurupalli, 2018). The 550 μl sample contained ~2 mM protein in a 20 mM sodium acetate, 50 mM NaCl, 1 mM NaN₃, 1 mM EDTA, 10% D₂O, pH 5.5 buffer.

NMR Experiments

The NMR experiments were carried out on Bruker Avance III HD (700 MHz) and Avance Neo (500 MHz) spectrometers. The 700 MHz spectrometer was equipped with a cryogenically cooled triple resonance probe while the 500 MHz spectrometer was equipped with a room temperature probe.

All the CEST experiments were performed at 700 MHz using the ¹⁵N CEST pulse sequence (Vallurupalli, Bouvignies, and Kay, 2012) in which the amide proton is decoupled from the amide ¹⁵N nucleus using 90_x240_y90_x inversion pulses (Levitt, 1982). At 42.9 °C, four CEST datasets were recorded using B₁ (T_EX; offset range) values of 53.6 (450 ms; ±1400 Hz), 107.2 (375 ms; ±1900 Hz), 200.3 (350 ms; ; ±1900 Hz) and 300.5 (300 ms; ±1900 Hz) Hz with the ¹⁵N carrier at 119.416 ppm. The spacing between adjacent B₁ offsets was 50 Hz for all four values of B₁. Each two-dimensional ¹⁵N-¹H^N correlation map was recorded with 18 complex points (sweep width of 1845 Hz) in the indirect dimension and four transients per FID leading to an acquisition time ~6 minutes per plane. The strength of the ¹⁵N B₁ field applied during the T_EX period was calibrated using the nutation method suggested by Guenneugues et al (Guenneugues, Berthault, and Desvaux, 1999). Very similar experimental parameters were used to record ¹⁵N CEST data at the other four temperatures (33.2, 35.5, 38.3 and 40.3 °C).

The transverse ¹⁵N-¹H^N dipole-dipole/¹⁵N CSA interference rate constants (η_xy) at different backbone amide sites in PSBD were obtained at 700 MHz (33.2, 35.5, 38.3, 40.3 and 42.9 °C) by measuring the decays (Bouguet-Bonnet, Mutzenhardt, and Canet, 2004) of the amide ¹⁵N TROSY and anti-TROSY components (Pervushin et al., 1997). Relaxation delays varied from 0 to 30 ms in steps of 5 ms with two repeats for error estimation. ¹⁵N TROSY and anti-TROSY decays were recorded in an interleaved manner in ~9 hours. Intrinsic transverse relaxation rates (R₂) were obtained from the (¹⁵N-¹H^N) dipole-dipole/(¹⁵N) CSA relaxation interference rate constant η_xy using the relation R₂ = kη_xy, with $\kappa =\sqrt{3}\frac{(4c^2+3d^2)}{12cd{P}_2(\cos \beta )}$ (Fushman, Tjandra, and Cowburn, 1998; Wang and Palmer, 2003). Here $c=\frac{{\gamma }_N{B}_0\Delta \sigma }{\sqrt{3}},d=\frac{-{\mu }_{0}h{\gamma }_H{\gamma }_N}{8{\pi }^2{r}_{NH}^3},$ h is the Planck’s constant, μ₀ the permittivity of free space, r_NH (=1.02 Å) is the N-H bond length, B₀ is the external magnetic field, Δσ (= -173 ppm) is the chemical shift anisotropy of the ¹⁵N site, β (=19.6°) is the angle between the N-H bond vector and the symmetry axis of the ¹⁵N chemical shift tensor while γ_H and γ_N are the gyromagnetic ratios of the ¹H and ¹⁵N nuclei respectively.

High-resolution amide ¹⁵N-¹H^N HSQC spectra to obtain the visible state peak positions (ϖ_Vis) in the ¹⁵N dimension were recorded at 700 MHz (33.2, 35.5, 38.3, 40.3 and 42.9 °C) using the experiment proposed by Skrynnikov et al. (Skrynnikov, Dahlquist, and Kay, 2002). The maximum evolution time in the indirect ¹⁵N dimension was 60 ms. Two ¹⁵N-¹H^N correlation maps were recorded at each temperature to estimate uncertainties in the peak positions. Each ¹⁵N-¹H^N correlation map was recorded in three hours.

¹⁵N and ¹H^N CPMG experiments (Gopalan, Hansen, and Vallurupalli, 2018) were carried out only at 42.9 °C at both 500 and 700 MHz. Amide ¹⁵N CPMG relaxation-dispersion data was recorded using a constant-time (Mulder et al., 2001) ¹⁵N TROSY-CPMG sequence (Loria, Rance, and Palmer, 1999b; Vallurupalli et al., 2007) with T_EX delays of 30 (500 MHz) and 20 (700 MHz) ms. Data was recorded for νCPMG values varying from 66.66 Hz (500 MHz)/100 Hz (700 MHz) to 1000 Hz. Amide ¹H^N CPMG data was recorded using a constant-time sequence (Ishima and Torchia, 2003) without a P-Element (Vallurupalli, Bouvignies, and Kay, 2011; Yuwen and Kay, 2019). T_EX was set to 20 ms with νCPMG varying from 100 Hz to 3000 Hz. To obtain a relaxation dispersion curve, ¹⁵N/(¹H^N) CPMG data was recorded at several (10-20) different ν_CPMG values with errors estimated based on a few repeat (2-5) measurements.

Data analysis

The NMRPipe suite of programs (Delaglio et al., 1995) was used for processing all the NMR data and SPARKY (Goddard and Kneller, 2008; Lee, Tonelli, and Markley, 2015) was used for visualization and to obtain peak centres. The software package PINT (Ahlner et al., 2013) was used to quantify the peak intensities in the various CEST/CPMG ¹⁵N-¹H^N correlation maps.

The ChemEx package (Bouvignies, 2012) that numerically integrates (Korzhnev et al., 2004) the Bloch-McConnell equations (McConnell, 1958) was used to estimate various global exchange (k_ex,FU, p_U) and residue specific parameters (ϖ_F, ϖ_U, R_1,F and R_2,F) from the data by minimising a standard ${\chi }^2={\displaystyle {\sum }_{i=1}^N\frac{{(m_i^{\mathit{\text{Calc}}}-m_i^{\mathit{\text{Exp}}})}^2}{{\sigma }_i^2}}$ function. Here $m_i^{Exp}$ is the experimental measure (i.e. peak intensity in the case of CPMG/CEST datasets or peak position in the case of the HSQC datasets), $m_i^{Calc}$ is the value calculated using the Bloch-McConnell equations and σ_i is the uncertainty in the experimentally measured value. The summation extends over all the experimental data used during the fitting process. Minimum uncertainties of 0.4% and 3 ppb were assumed for the ¹⁵N CEST intensities and visible peak positions (ϖ_Vis) respectively. The two-state (F⇌U) Bloch-McConnell equations were constructed assuming R_2,U = R_2,F/2 and R_1,U = R_1,F via previously published procedures (Korzhnev et al., 2004; Vallurupalli, Bouvignies, and Kay, 2012). Here R_2,U was constrained to be R_2,F/2 (Farrow et al., 1995) as a different dip is not observed for the U state in the CEST intensity profiles (Tiwari et al., 2021). Intensities in the case of the CEST/CPMG experiments were calculated by numerically integrating the Bloch-McConnell equations for the duration of the T_EX period (Korzhnev et al., 2004; Vallurupalli, Bouvignies, and Kay, 2012). Visible peak-positions (ϖ_Vis) were calculated from the eigenvalues of the Bloch-McConnell equations (Anet and Basus, 1978; Skrynnikov, Dahlquist, and Kay, 2002). Uncertainties in the ¹⁵N CEST intensities were estimated from the scatter in the flat parts of the intensity profile (Vallurupalli, Bouvignies, and Kay, 2012). Due to the broad dip sizes in the ¹⁵N CEST intensity profiles, only the B₁ ~50 Hz profiles contained flat portions. Hence uncertainties were estimated only from the B₁ ~50 Hz profiles and these same values were used to analyse the CEST profiles recorded with higher B₁ values.

¹⁵N CEST data from a select set (Set 1) of 18 residues was analysed globally (common k_ex,FU and p_U). These 18 residues (I3, A4, V8, R9, A12, R13, K15, D18, I19, R20, L21, Q23, G24, G29, V31, D37, L40 and L41) had CEST derived ¹⁵N |Δϖ_FU| values ≥ 3 ppm at 42.9 °C. The same set of 18 residues was used for the global analysis of ¹⁵N CPMG data at 42.9 °C and ¹⁵N CEST data at all the other temperatures. ¹H^N CPMG data from 12 residues (I3, M5, V8, R13, K15, G16, V22, Q23, R30, K33, I36, D37) for which R_2,eff (100 Hz) – R_2,eff (3000 Hz) ≥ 5 s^-1 at both 500 and 700 MHz was analysed globally using a two-state exchange model. Residue specific Δϖ_FU values for the other sites (other than Fig 1) were obtained by fixing the k_ex,FU and p_U values to those obtained from the global analysis of the above specified subset(s) of residues. In some cases the analysis was performed on a smaller set (Set 2) of 14 residues (I3, A4, V8, R9, R13, D18, R20, Q23, G24, G29, V31, D37, L40 and L41) that had CEST derived ¹⁵N |Δϖ_FU| values between 3 and 5 ppm at 42.9 °C. Uncertainties in the best-fit exchange parameters were obtained using a standard bootstrap procedure (Choy et al., 2005; Press et al., 1992) with 250 trials.

Fig. 1. Amide ¹⁵N CEST detects sparsely populated states with ~100 μs lifetimes.

a) At 42.9 °C the folded structure of PSBD [PDB: 1W3D (Allen et al., 2005)] that consists of three helices exchanges with the unfolded (U) state populated to < 5% on the ~100 μs timescale. b) The 700 MHz amide ¹⁵N-¹H^N correlation map of U-[¹⁵N] PSBD (42.9 °C) is well resolved and contains only correlations arising from the folded (F) state. Peaks are labeled according to the residue from which they arise. It should be noted that peak-positions do not strictly correspond to the F state but are shifted towards the U state as the system is in fast exchange between the F and U states. c) ¹⁵N CEST intensity profiles from A12 and R20. Experimental data is represented using filled red circles and the blue line is calculated using best-fit parameters (p_U = 3.9% and k_ex,FU = 11363 s^-1). d) Large variations are observed in the ¹⁵N CEST derived Δϖ_FU values along the length of the sequence. e) The ¹⁵N CEST derived Δϖ_FU values are in excellent agreement with the predicted Δϖ_FU values. ϖ_U values to calculate Δϖ_FU were obtained using the program POTENCI (Nielsen and Mulder, 2018).

Kinetic and thermodynamic parameters ($\Delta H_{UF}^{*},\Delta S_{UF}^{*},$ ΔH_UF and ΔS_UF) for the PSBD folding process were obtained by analysing the temperature dependence of the folding (k_UF) and unfolding (k_FU) rate constants as described previously (Vallurupalli et al., 2016) using modified Kramers-Arrhenius equations, $k_{UF}(T)=\frac{k_0}{\eta (T)/{\eta }_0}{e}^{-\frac{(\Delta H_{UF}^{*}-T{\text{ΔS}}_{UF}^{*})}{RT}}$ and $k_{FU}(T)=\frac{k_0}{\eta (T)/{\eta }_0}{e}^{-\frac{(\Delta H_{UF}^{*}-\Delta H_{UF}-T(\Delta S_{UF}^{*}-\Delta S_{UF}))}{RT}}$ (Ansari et al., 1992; Hagen, 2010; Sekhar, Vallurupalli, and Kay, 2012). $\Delta H_{UF}^{*}\text{and}\Delta S_{UF}^{*}$ are respectively the activation enthalpy and activation entropy for the folding reaction. ΔH_UF and ΔS_UF are respectively the change in enthalpy and entropy upon folding. R is the universal gas constant, η(T) is the viscosity at temperature T, while η₀ is (1 cP) the viscosity of water at a reference temperature of 293.15 K. k₀ was set to 2.27×10⁶ s^-1 (at all temperatures) according to the formula suggested by Eaton (Eaton, 2021) for the folding speed-limit of a 44 residue protein. A Monte Carlo procedure (Press et al., 1992) with 100 trials was used to estimate the uncertainties in the estimates of $\Delta H_{UF}^{*},\Delta S_{UF}^{*},$ ΔH_UF and ΔS_UF.

Results and Discussion

¹⁵N CEST detects exchange occurring on the 100 μs timescale

The 44 residue PSBD from the pyruvate dehydrogenase multienzyme complex of Bacillus stearothermophilus is a largely helical protein (Allen et al., 2005) that folds on the ~10-100 μs timescale (Fig 1a). PSBD has served as a model system to understand protein folding and its folding has been extensively investigated by several techniques (Ferguson et al., 2005; Spector and Raleigh, 1999; Vugmeyster et al., 2000). At 42.9 °C the amide ¹⁵N-¹H^N correlation map is well resolved containing peaks arising from the native folded state (Fig 1b). The ¹⁵N CEST profiles recorded using a U-[¹⁵N] PSBD sample contained a single dip near the visible state resonances (Fig 1c). However a model in which only the native state is populated at 42.9 °C did not account $({\chi }_{\mathit{\text{red}}}^2=2)$ for the ¹⁵N CEST profiles (B₁ values of 53.6, 107.2, 200.3 and 300.5 Hz) obtained from 33 different residues. The ¹⁵N CEST profiles were however accounted for $({\chi }_{\mathit{\text{red}}}^2=0.77)$ by a global two-state exchange model in which the folded (F) state exchanges with an ‘invisible’ unfolded state (U) with best-fit p_U = 3.9% and k_ex,FU = 11363 s^-1. There are large changes in chemical shift between major state and the minor state along the sequence (Fig. 1d) and the minor state chemical shifts obtained from the analysis of the CEST data are in very good agreement (Fig. 1e, rmsd 0.91 ppm) with predicted U state shifts (Nielsen and Mulder, 2018) confirming that the minor state is indeed the unfolded state. In the discussion that follows the F state will refer to the major state and the U state to the minor state.

To test if the exchange parameters obtained from the analysis of the ¹⁵N CEST data are reliable we analysed data from a select set of 18 residues for which |Δϖ_FU| ≥ 3 ppm (see materials and methods). As before the two-state model accounted for the ¹⁵N CEST data $({\chi }_{red}^2=0.76)$ and although reasonably precise estimates of p_U (4.0 ± 0.32%) and k_ex,FU (11317 ± 376) s^-1 were obtained, the ${\chi }_{red}^2$ versus k_ex,FU and ${\chi }_{red}^2$ versus p_U plots (black line in Fig 2a,b) are quite flat. In our experience these shallow or non-existent minima can lead to spurious “precise” solutions due to some unaccounted artifacts/features in the data that the model attempts to satisfy. Consequently to rule out spurious solutions and determine reliable exchange parameters it is important to obtain more pronounced global minima in the ${\chi }_{red}^2$ versus k_ex,FU and ${\chi }_{red}^2$ versus p_U plots. This can be achieved by constraining the fitting parameters to reasonable values (Palmer and Koss, 2019) or by judiciously including in the fitting procedure additional experimental data that complements the existing data (Bouvignies et al., 2011; Farber, Slager, and Mittermaier, 2012; Gopalan and Vallurupalli, 2018; Igumenova et al., 2007; Korzhnev et al., 2005; Mulder et al., 1999; Neudecker, Korzhnev, and Kay, 2006; Vallurupalli, Bouvignies, and Kay, 2011).

Fig. 2.

Experimentally derived constraints (R_2,const) on R_2,F values and the inclusion of accurate ϖ_Vis shifts in the analysis of amide ¹⁵N CEST data leads to pronounced minima in the (a) ${\chi }_{red}^2$ versus k_ex,FU and (b) ${\chi }_{red}^2$ versus p_U plots. In (a) and (b) best-fit calculations performed using only ¹⁵N CEST data is shown in black, calculations performed using R_2,F constraints (R_2,const) are shown in blue, calculations performed with the addition of only ϖ_Vis shifts are shown in green and calculations performed using both R_2,F constraints and ϖ_Vis shifts are shown in red. (c) Root mean square deviation (rmsd) between the measured ϖ_Vis shifts and those calculated from parameters obtained from best-fit calculations using ¹⁵N CEST data and R_2,F constraints (blue in (b)) plotted as a function of p_U. (d) Variation of the ${\chi }_{red}^2$ versus p_U plots as a function of precision in the ϖ_Vis shifts. Best-fit calculations were performed using ¹⁵N CEST with R_2,F constraints while artificially increasing the uncertainty (σϖ_Vis) on the measured ϖ_Vis shifts from 3 to 50 ppb. For reference the results of the optimization performed without ϖ_Vis shifts but including R_2,F constraints (blue in b) is shown in blue. The calculations were performed by globally analyzing data from 18 residues (Set 1) for which |Δϖ_FU| ≥ 3 ppm.

Constraints on the major-state transverse relaxation rates leads to more reliable exchange parameters

The best-fit procedure to obtain exchange parameters from the ¹⁵N CEST data optimises residue specific parameters ϖ_F, ϖ_U, R_1,F and R_2,F in addition to the global exchange parameters (k_ex,FU and p_U). Here R_1,i and R_2,i respectively refer to the longitudinal and intrinsic transverse relaxation rates of the spin of interest in state i (i ∈ F,U). To test if any of the residue-specific parameters might be compensating for improper k_ex,FU and p_U values, we examined how the best-fit residue-specific parameters varied as a function of k_ex,FU and p_U (Fig S1). (In these fits, constraints R_2,F ≥ 0 s^-1 and R_2,U = R_2,F/2 were imposed.) For the ~4.7 kDa PSBD, we expect the amide ¹⁵N R_2,F to be around ~3 to 4 s^-1 at 42.9 °C. However plots of the best-fit R_2,F values versus k_ex,FU (Fig S1a) and p_U (Fig S1b) show that the fitted R_2,F values vary significantly as a function of k_ex,FU and p_U. They can be very large (> 8 s^-1) when k_ex,FU and p_U are smaller than the ‘right’ values (grey line Fig S1a) and very small (~0 s^-1) when k_ex,FU and p_U are greater than the ‘right’ values. (For the discussion here, k_ex,FU = 11739 s^-1 and p_U = 4.3% obtained later using more data and restraints [Table 1] are considered to be the ‘right’ exchange parameters.) To try and rationalise these trends in the fitted R_2,F values, we tried to understand how various parameters affect the shape of the CEST intensity profile that can be approximated (Palmer, 2014; Palmer and Koss, 2019) as:

$$I/I_0\approx {\cos }^2\theta e^{-R_{1\rho }{T}_{EX}}$$ (1)

with

$$R_{1\rho }({\omega }_1,{\omega }_{RF})=R_1{\cos }^2\theta +(R_2+R_{ex}({\omega }_1,{\omega }_{RF})){\sin }^2\theta$$ (2)

Table 1.

Summary of various parameters obtained from the global two-state (F⇌U) modeling of ¹⁵N CEST recorded on PSBD at various temperatures. Set 1 refers to 18 residues with |Δϖ_FU| ≥ 3 ppm at 42.9 °C. Set 2 refers to 14 residues for which: 3 ppm ≤ |Δϖ_FU| ≤ 5 ppm. See materials and methods for a list of the residues contained in sets 1 and 2. Columns titled R_2,const and ϖ_Vis shifts are respectively used to indicate if experimentally derived constraints on R_2,F values (R_2,const) or ϖ_Vis shifts were included in the analysis.

S No	Temperature (°C)	Residue Set	R2,const	ϖVis	kex,FU (s-1)	pU (%)	${\chi }_{red}^2$
1	42.9	Set 1	No	No	11317 ± 376	4.00 ± 0.32	0.76
2	42.9	Set 1	Yes	No	11688 ± 142	4.43 ± 0.20	0.77
3	42.9	Set 1	No	Yes	10592 ± 387	3.23 ± 0.31	0.77
4	42.9	Set 1	Yes	Yes	11739 ± 147	4.30 ± 0.17	0.78
5	42.9	Set 2	No	No	11189 ± 421	3.61 ± 0.38	0.72
6	42.9	Set 2	Yes	No	11654 ± 179	4.19 ± 0.21	0.72
7	42.9	Set 2	No	Yes	11458 ± 454	2.83 ± 0.37	0.73
8	42.9	Set 2	Yes	Yes	11788 ± 194	4.14 ± 0.22	0.74
9	40.3	Set 1	Yes	Yes	10788 ± 183	2.88 ± 0.14	0.86
10	38.3	Set 1	Yes	Yes	11046 ± 240	2.06 ± 0.12	0.94
11	35.5	Set 1	Yes	Yes	9974 ± 262	1.10 ± 0.09	0.99
12	33.2	Set 1	Yes	Yes	10331 ± 379	0.73 ± 0.09	1.05

For simplicity here we have assumed that both states have the same longitudinal (R₁) and intrinsic transverse (R₂) relaxation rates. Then the exchange contribution to the transverse relaxation (R_ex) of the visible state can be approximated as (Palmer, 2014; Palmer and Koss, 2019; Trott and Palmer, 2002):

$$R_{ex}({\omega }_1,{\omega }_{RF})\approx \frac{p_F{p}_U\Delta {\omega }_{FU}^2}{k_{ex,FU}}\left[\frac{k_{ex,FU}^2}{k_{ex,FU}^2+\frac{{\omega }_{F,eff}^2{\omega }_{U,eff}^2}{{\omega }_{eff}^2}}\right]$$ (3)

Here ${\omega }_{i,eff}^2=({\text{Ω}}_i^2+{\omega }_1^2),$ ω₁ = 2πB₁, Ω_i = ω_i – ω_RF is the offset (rad/s) of the nucleus of interest in state i (i ∈ F,U) from the frequency (ω_RF) at which the B₁ field is applied, ${\omega }_{eff}^2=({\overline{\Omega }}^2+{\omega }_1^2)$ with $\overline{\Omega }=p_F{\omega }_F+p_U{\omega }_U-{\omega }_{RF}$ and $\tan \theta ={\omega }_1/\overline{\Omega }\text{.}$

It is clear from equation 3 that R_ex depends on the strength of B₁ field, the offset (ω_RF) at which it is applied, exchange parameters (k_ex,FU, p_U) and the resonance frequencies of the spin of interest in the major (ω_F) and minor (ω_U) states. Hence mis-setting k_ex,FU or p_U can lead to erroneous R_ex values that can be compensated for by adjusting the value of R₂ (Equation 2) in a way that the shape of the CEST intensity profile remains nearly the same (Vugmeyster et al., 2000). Thus, constraining the R₂ rates to their correct values can lead to more accurate estimates of the best-fit exchange parameters. R₁ on the other hand is well defined from the flat parts of the CEST profiles. It is useful to note that in the case of slow-exchange unlike the case of intermediate/fast exchange being studied here, the CEST intensity profile will contain separate (relatively narrow) dips for the major and minor states and accurate intrinsic transverse relaxation rates for the two states can be obtained from the analysis of the CEST data (Vallurupalli, Bouvignies, and Kay, 2012) and there may be no need to additionally constrain them. In fact CEST experiments have been used to measure intrinsic R₂ rates both in the absence (Bain, Ho, and Martin, 1981) and presence (Gu et al., 2016) of exchange. In the case of fast exchange for reasons similar to those discussed above, R₂ rates have been constrained while analyzing CPMG (O'Connell et al., 2009) and R_1ρ (Vugmeyster et al., 2000) data. The potential power of R₂ constraints in the context of CEST experiments was recently illustrated in a study of the folding of the A39G FF domain using ¹⁵N CEST experiments at 600 and 1000 MHz. The R₂ values of the minor states were linked to the R₂ values of the major state (that were fit) and this resulted in the detection of exchange occurring at ~8,500 s^-1 between two minor states populated to just ~1% and ~0.34% respectively in a four state system (Tiwari et al., 2021). Here we are proposing doing the same for the major state to study rapid exchange between the major and a minor state.

The transverse relaxation rate of the ¹⁵N nucleus in an amide ¹⁵N-¹H^N spin-system is dominated by its chemical-shift anisotropy (CSA) and the dipole-dipole interaction with the attached proton and reasonably accurate estimates of the intrinsic (exchange free) ¹⁵N R₂ values can be very conveniently obtained from the transverse (¹⁵N-¹H^N) dipole-dipole/(¹⁵N) CSA relaxation interference rate constant (η_xy) that is insensitive to exchange (Fushman, Tjandra, and Cowburn, 1998; Wang and Palmer, 2003). Other approaches have also been proposed to obtain the exchange free intrinsic transverse relaxation rates of the visible state (Hansen et al., 2007; Phan, Boyd, and Campbell, 1996). Here we have confirmed that the ¹⁵N R₂ values obtained from η_xy values are in reasonable agreement (~±10%, maximum deviation of ~30%) with those obtained from ¹⁵N CEST data by carrying out measurements on ¹⁵N enriched samples of T4 lysozyme (Fig. S2). Next we obtained estimates of the R_2,F values from η_xy measurements on the PSBD sample at 42.9 °C (Fig. S3) and constrained the fitted R_2,F values to be within±50% of the estimates obtained from the η_xy measurements. The constraints did not alter the quality of the fits $({\chi }_{\mathit{\text{red}}}^2=0.77)$ but lead to slightly more precise estimates of exchange parameters, p_U = 4.43 ± 0.20% and k_ex,FU = 11688 ± 142 s^-1. The ${\chi }_{red}^2$ versus k_ex,FU and ${\chi }_{red}^2$ versus p_U plots with the constraints on R_2,F values are shown using blue lines in figures 2a and 2b respectively. It is clear from the ${\chi }_{red}^2$ versus k_ex,_FU plot that constraints on the R_2,F values does not allow for solutions (compare the blue and black lines in Fig. 2a) with low k_ex,FU values leading to a clear minimum at ~11,688 s^-1. Constraints on the R_2,F values also rule out solutions (compare the blue and black lines in Fig. 2b) with low p_U values but still allow for solutions with large p_U values. Thus unlike in the ${\chi }_{red}^2$ versus k_ex,FU plot we still do not have a clear minimum in the ${\chi }_{red}^2$ versus p_U plot.

Visible state peak-positions can also lead to more reliable exchange parameters

When exchange is fast (k_ex,FU ≫ |Δω_FU|), it is difficult to get independent estimates of the minor state population (p_U) and the chemical shift difference (Δϖ_FU) from relaxation data as they can be varied in a correlated manner to reproduce the measured relaxation rates (Luz and Meiboom, 1963; Palmer, 2004). This is evident from equation 3 which shows that changes in p_U can be compensated by adjusting Δω_FU such that $p_F{p}_U\Delta {\omega }_{FU}^2$ remains constant. It is clear from Figure S1c that large p_U values are accommodated by the ¹⁵N CEST data here (Fig 1b blue line) by scaling Δω_FU such that $p_F{p}_U\Delta {\omega }_{FU}^2(\text{or}{p}_F{p}_U\Delta {\varpi }_{FU}^2)$ remains largely constant. To break this correlation between p_U and Δω_FU we need to include in the fitting procedure experimental data that has a different dependence on p_U and Δω_FU. The exchange induced shift (δ_ex; rad/s) of the dominant visible state peak is one such parameter (Palmer, 2004; Vallurupalli, Bouvignies, and Kay, 2011). When p_U ≪ 1 we have (Palmer, 2004; Swift and Connick, 1962; Vallurupalli, Bouvignies, and Kay, 2011):

$${\delta }_{ex}=\frac{p_U\Delta {\omega }_{FU}}{1+{(\frac{\Delta {\omega }_{FU}}{k_{ex,FU}})}^2}$$ (4)

Hence when k_ex,FU ≫ |Δω_FU| equation 4 simplifies to:

$${\delta }_{ex}\approx p_U\Delta {\omega }_{FU}(1-(\frac{\Delta {\omega }_{FU}}{k_{ex,FU}}{)}^2)$$ (5)

It is clear from equations 3 and 5, that R_ex and δ_ex have different dependencies on p_U and Δω_FU in the fast exchange limit (and p_U ≪ 1). As $R_{ex}\propto p_U\Delta {\omega }_{FU}^2$ and δ_ex ∝ p_UΔω_FU it will not be possible to compensate for incorrect p_U values by scaling Δω_FU to obtain the correct R_ex and δ_ex values simultaneously. δ_ex cannot be directly measured, but the visible peak position ω_Vis = ω_F + δ_ex can be measured very accurately from ¹H^N-¹⁵N correlation maps. In figure 2c, root mean square deviation between the experimental visible state peak position in ppm (ϖ_Vis) and visible state peak position calculated (ϖ_Vis,calc) from parameters obtained from the best-fit procedure that included experimentally derived constraints on the fitted R_2,F values is shown for different p_U values. When p_U ~4% the deviation between the calculated and measured values is less than 30 ppb (~2 Hz at 700 MHz) but increases to more than 60 ppb when p_U in increased to 10% (Fig 2c). From equations 1–3 it is clear that the shape of the CEST intensity profile is a complicated function various exchange parameters and should contain information regarding ϖ_Vis which is probably why the ϖ_Vis values can be predicted reasonably well (~30 ppb) from the CEST derived best-fit parameters. However better estimates of ϖ_Vis values will probably not be available from the CEST data alone as the dips are on the order of a (few) ppm wide (Fig 1c). As ϖ_Vis can be measured very precisely (< 3 ppb) from ¹⁵N-¹H^N correlation maps, a combined analysis of experimentally measured ¹⁵N ϖ_Vis values along with ¹⁵N CEST data could make spurious solutions with large p_U values unlikely. Including the visible state peak-positions in fitting process along with the constraints on the R_2,F values does not alter the quality of the fits $({\chi }_{red}^2=0.78)$ or the estimates of exchange parameters, p_U = 4.3 ± 0.17% and k_ex,FU = 11739 ± 147 s^-1 but leads to a pronounced minimum in the ${\chi }_{red}^2$ versus p_U plot as two-state solutions with higher p_U values can no longer satisfy the data (Fig 2b; red line). Including just the visible peak positions without constraints on the R_2,F values during the fitting processes also rules out solutions with large p_U values (compare the green and blue lines in Fig 2b). Consistent with our earlier speculation that the high precision of the measured ϖ_Vis values is critical to improving the shape of the ${\chi }_{red}^2$ versus p_U plot, we find that the well defined minimum starts to vanish when the uncertainty in the ϖ_Vis values is artificially increased from 3 to 30 ppb (Fig 2d). Even though the ϖ_Vis values do not lead to a quantitative improvement here, they rule out solutions with large p_U values confirming that the well-defined best-fit exchange parameters obtained without any constraints accurately reflect the kinetics in the sample and did not arise due to some artefacts in the data. It is worth noting that information from ϖ_Vis values is often implicitly included while analysing CEST data. This is done by either insisting that the major-state resonance is at ϖ_Vis in the case of slow exchange or that the population weighted chemical shift is at ϖ_Vis in the case of fast-exchange (Rangadurai, Shi, and Al-Hashimi, 2020; Vallurupalli, Bouvignies, and Kay, 2012).

To test if the p_U and k_ex,FU parameters determined from the above analysis of the ¹⁵N CEST data are accurate, we compared them to exchange parameters obtained from ¹H^N and ¹⁵N CPMG data (Fig 3a,b) recorded on the same sample at 500 and 700 MHz. The CPMG data is consistent with a two-state exchange processes $({\chi }_{\mathit{\text{red}}}^2=0.83)$ and the ${\chi }_{red}^2$ versus k_ex,FU plot obtained from the analysis of the CPMG data has a clear minimum with a best-fit k_ex,FU = 10987 ± 275 s^-1 in reasonable agreement with the CEST derived value of 11739 ± 147 s^-1 (Fig 3c). Based on previous ¹H^N and ¹⁵N CPMG studies of PSBD folding (Gopalan and Vallurupalli, 2018; Gopalan et al., 2018) we did not expect to obtain estimates of p_U from the ¹H^N and ¹⁵N CPMG data alone as the system is in fast exchange which means that exchange models with incorrect (larger) p_U values can fit the data by scaling the fitted Δϖ_FU values by $\sqrt{p_{U,Correct}/p_{U,Fitted}}$ (Gopalan and Vallurupalli, 2018; Luz and Meiboom, 1963; Vallurupalli, Bouvignies, and Kay, 2011). Here p_U,correct and p_U,Fitted refer to the correct and fitted p_U values respectively. Hence if the CEST derived p_U value is correct there should be a good correlation with a slope of ~1 between the CPMG derived Δϖ_FU values with p_U fixed to the CEST derived value of 4.3% and those reported previously by simultaneously analysing ¹H^N, ¹⁵N, methyl ¹H TQ CPMG data along with ¹H^N/¹⁵N H(S/M)QC & methyl ¹H SQ/DQ/TQ shift data (Vallurupalli, Bouvignies, and Kay, 2011; Yuwen, Vallurupalli, and Kay, 2016). As expected the ${\chi }_{red}^2$ versus p_U plot obtained from an analysis of the ¹H^N and ¹⁵N CPMG data does not contain a pronounced minimum (Fig 3d). However, there is a small minimum at p_U = 2.21%, that turns out to be a spurious minimum of the kind that we are worried about. When the ¹H^N & ¹⁵N CPMG data was analysed with p_U set to the ¹⁵N CEST derived value of 4.3% the slope between the CPMG derived ¹H^N |Δϖ_FU| and ¹⁵N Δϖ_FU values and the previously reported ¹H^N |Δϖ_FU| and ¹⁵N Δϖ_FU values is 0.92 (Fig 3e) showing that the p_U of 4.3% obtained from the ¹⁵N CEST data is correct (0.92² ≈ 0.85) to within 20%, assuming that the previously reported Δϖ_FU values are accurate. On the other hand there is a slope of 0.69 between the CPMG derived ¹H^N |Δϖ_FU| and ¹⁵N Δϖ_FU values with p_U = 2.21% and the previously reported Δϖ_FU values (Fig 3f) showing that the CPMG derived p_U of 2.21% is about half $({0.69}^2=0.48\approx (\frac{2.21\text{\%}}{4.3\text{\%}}))$ of the ‘correct’ value confirming that the CPMG derived p_U of 2.21% is inaccurate and arises from a spurious minimum.

Fig. 3.

CPMG data is consistent with exchange parameters obtained from the analysis of ¹⁵N CEST data. Representative ¹H^N (a) and ¹⁵N (b) CPMG relaxation dispersion profiles recorded at 500 and 700 MHz. The experimental data is represented using filled circles while the line is calculated using the best-fit parameters (p_U = 2.21% and k_ex,FU = 10987 s^-1). (c) ${\chi }_{red}^2$ versus k_ex,FU and (d) ${\chi }_{red}^2$ versus p_U plots obtained using ¹H^N & ¹⁵N CPMG data (purple) and ¹⁵N CEST data (red) along with ϖ_Vis shifts and constraints on R_2,F values. The minima in the ${\chi }_{red}^2$ versus k_ex,FU curves (c) are consistent with one another, while the ${\chi }_{red}^2$ versus p_U curve derived from the CPMG data does not have a convincing minimum except for a small spurious one at 2.21% (d). Comparison between the reported ¹H^N |Δϖ_FU| and ¹⁵N Δϖ_FU values and those obtained here by setting p_U to 4.3% (e) and 2.21% (f). In (e) and (f) the combined slopes were calculated for ¹H^N |Δϖ_FU| and ¹⁵N Δϖ_FU values by multiplying the ¹H^N |Δϖ_FU| values by 10.

The effect of including R₂ constraints and ϖ_Vis values during the analysis of ¹⁵N CEST data becomes even more apparent (Fig 4a,b) when we restrict the analysis to a select set of 14 residues (Set 2) that have |Δϖ_FU| values between 3 and 5 ppm (Fig 4). Without any restraints p_U = 3.61 ± 0.38% and k_ex,FU = 11189 ± 421 adopt (Fig 4c,d,e) a slightly wide-range of values in 250 boot-strap trials (Fig 4e), but are a little better defined (p_U = 4.14 ± 0.22% and k_ex,FU = 11788 ± 194) with the restraints (Fig 4). Here too leaving out the ϖ_Vis values during the fitting process does not affect the precision of the exchange parameters (p_U = 4.19 ± 0.21% and k_ex,FU = 11654 ± 179 s^-1) but a clear minimum appears in the ${\chi }_{red}^2$ versus p_U plot only when the ϖ_Vis values are included in the fitting process (Fig 4b) showing that a convincing minimum can appear in ${\chi }_{red}^2$ versus p_U or the ${\chi }_{red}^2$ versus k_ex,FU plots even in the case of relatively fast exchange, k_ex,FU/|Δω_FU| ~5.

Fig. 4.

Experimentally derived constraints (R_2,const) on R_2,F values and the inclusion of accurate ϖ_Vis shifts during data analysis has a large effect on the exchange parameters extracted by analysing ¹⁵N CEST data from the Set 2 residues (5 ppm ≥ |Δϖ_FU| ≥ 3 ppm). (a) and (b) were computed exactly as in Fig 2a and Fig 2b respectively, but using only the 14 Set 2 residues. Distribution of k_ex,FU (c) and p_U (d) values obtained from 250 bootstrap trials using only ¹⁵N CEST data (black) and using both R_2,F constraints and ϖ_Vis shifts in addition to ¹⁵N CEST data (red). e) Scatter plot showing different k_ex,FU and p_U values obtained in the bootstrap trials.

Kinetics and Thermodynamics of PSBD folding

To obtain insights into the rapid folding of PSBD under the conditions used here, we performed amide ¹⁵N CEST experiments, η_xy and ϖ_Vis measurements at four additional temperatures 33.2, 35.5, 38.3, and 40.3 °C and analysed them using two-state (F⇌U) exchange model as described above to obtain exchange parameters at these additional temperatures (Table 1). We find that k_ex,FU is not temperature dependent (Fig 5a) while p_U increases with temperature (Fig 5b, c) which means that the folding rate constant (k_UF) is largely independent of temperature while the unfolding rate constant (k_FU) increases with temperature (Fig 5d). Such behaviour has been observed for several fast-folding proteins including PSBD (Bahar, Jernigan, and Dill, 2017; Spector and Raleigh, 1999) and predicted by the Zwanzig, Szabo, Bagchi (ZSB) model for protein folding (Zwanzig, 1995; Zwanzig, Szabo, and Bagchi, 1992). An Arrhenius analysis of rates leads to ΔH_FU = 148.5 ± 9.7 kJ/mol, ΔS_FU = 444 ± 31 J/mol·K, $\Delta H_{FU}^{*}=143.6\pm 9.7\text{kJ}/\text{mol}$ and $\Delta S_{FU}^{*}=380.6\pm 30.9\text{J}/\text{mol}\cdot \text{K}$ (Fig 5c, d, e). Here ΔH_FU and ΔS_FU are differences in the enthalpy and entropy between the U and F states respectively. $\Delta H_{FU}^{*}\text{and}\text{Δ}{S}_{FU}^{*}$ are respectively the activation enthalpy and entropy for the FU process leading to activation parameters $\Delta H_{UF}^{*}=-4.9\pm 2.4\text{kJ}/\text{mol}$ and $\Delta S_{UF}^{*}=-63.8\pm 7.6\text{J}/\text{mol}\cdot \text{K}$ for the folding (UF) process. While the activation free energy $\Delta G_{FU}^{*}$ for the unfolding process is 23.3 kJ/mol (8.9 RT) at 316.05 K, the activation free energy for the folding process $\Delta G_{UF}^{*}$ is a modest 15.2 kJ/mol (5.8 RT). Hence as the transition (T) state is only slightly more unstable (higher G) than the U state and PSBD is able to fold rapidly because the folding barrier is small (5.8 RT). Compared to the U state, the T state is is more ordered $(\text{-ve}\Delta S_{UF}^{*})$ and also has favourable interactions $(\text{-ve}\Delta H_{UF}^{*})$ but the favourable interactions do not overcome $(\Delta G_{UF}^{*}>0)$ the loss of entropy due to ordering leading to the small folding barrier in line with the ZSB model (Zwanzig, 1995; Zwanzig, Szabo, and Bagchi, 1992).

Fig. 5.

Understanding the PSBD folding/unfolding reactions. Distribution of k_ex,FU (a) and p_U (b) values obtained from 250 bootstrap trials at various temperatures (33.2 to 42.9 °C). The exchange parameters (Table 1) were obtained from ¹⁵N CEST data supplemented with R_2,F constraints and ϖ_Vis shifts. van’t Hoff (c) and Arrhenius (d) plots of exchange parameters obtained at different temperatures. As described in Materials and Methods, the thermodynamic (ΔH_UF and ΔS_UF) and activation ($\Delta H_{UF}^{*}\text{and}\Delta S_{UF}^{*}$) parameters were obtained using the data shown in the Arrhenius plots (d). The van’t Hoff plot (c) is shown here to illustrate how p_U increases with temperature and the best-fit ΔH_UF and ΔS_UF values obtained in (d) were used to generate the blue line. e) Decomposition of the Gibbs free energy (G) for the U and T states into its enthalpic (H), and entropic (TS) components. H and S are set to 0 for state F. The reference temperature in (e) is 316.05 K.

Concluding Remarks

Here we have shown that imposing experimentally derived constraints on the major-state transverse relaxation rate and including the visible-state peak position during the analysis of amide ¹⁵N CEST data acquired with very modest B₁ fields (~50 to ~300 Hz) results in reliable and more precise estimates of exchange parameters even when exchange rates are on the order of 10,000 s^-1 that lies in the fast exchange limit with k_ex/|Δω| values of ~5 (here for Set 2). Relatively fast processes can also be studied using a combination of amide ¹H^N/¹⁵N CPMG experiments in combination with H(S/M)QC shifts (Vallurupalli, Bouvignies, and Kay, 2011). Even though ¹⁵N CEST and ¹H^N/¹⁵N CPMG & H(S/M)QC data are recorded from the same amide sites, large (useful) shift differences between ¹⁵N-¹H^N HSQC and HMQC spectra are observed only for sites where ¹H^N and ¹⁵N |Δω| values are large and similar, reducing the efficacy of this strategy compared to the ¹⁵N CEST based strategy that works so long as the ¹⁵N |Δω| values are large. Very fast exchange (~25,000 s^-1) can also be studied using amide ¹⁵N E-CPMG and high-power R_1,ρ experiments but they do not directly provide estimates of the minor state populations and chemical shifts (Ban et al., 2012; Reddy et al., 2018). These various approaches can in principle be combined with the ¹⁵N CEST based strategy presented here to obtain more accurate estimates of the exchange parameters. Calculations (Fig. S4) also suggest that the strategy presented here of constraining R_2,F values along with the inclusion of accurate ϖ_Vis shifts while analysing ¹⁵N CEST data will work for medium sized proteins (R_2,F = 15 s^-1, ~15 kDa at 25 °C). It may be possible to extend the strategies presented here in the context of the amide ¹⁵N-¹H^N system to study exchange at various ¹³C sites in proteins using site-specifically ¹³C enriched samples (Goto et al., 1999; Hansen et al., 2008; Lundstrom, Lin, and Kay, 2009; Lundstrom et al., 2007; Lundstrom et al., 2009) that are free from the deleterious effects of carbon-carbon J couplings (Bouvignies, Vallurupalli, and Kay, 2014; Vallurupalli, Bouvignies, and Kay, 2013; Zhou and Yang, 2014).

As the utility of the amide ¹⁵N CEST experiments to study protein dynamics occurring on the ~10^-1 to ~10^-2 second timescale is well established, the developments presented here show that proteins dynamics occurring over the entire ~10^-1 to ~10^-4 second timescale can be studied using simple amide ¹⁵N CEST experiments performed on a single uniformly ¹⁵N enriched sample.